The Cross Products tab in the Text Editor allows you to create new intersection relationships for a term of interest (that is, the current sub-selection) based on a specified genus and collection of differentia. The Cross Products tab does not create any new terms. To create cross-product relationships for an existing term, load the term into the Text Editor panel by selecting it. To create a brand new term with a cross product definition, first create the new term, and then specify the cross product definition as described below.
The text editor has a tab called "Cross Products":

This tab assumes that your cross products conform to the genus/differentia method of designating cross products.
The genus term is listed at the top of the Cross Products tab. Differentia terms are listed in the "Discriminating Relations" section. Each line of the Discriminating Relations section represents a relationship to a term. The dropdown box on the left contains the relationship type, and the text box immediately to the right contains the target (parent) term of this relationship.
To the right of each term name in the Cross Products editor is a button marked with a fuzzy light blue circle. Clicking this button causes that term to be selected in the Ontology Tree Editor panel (in place of the currently shown term). The fuzzy circle buttons are intended for use while browsing, not editing. Clicking one of these buttons will usually cause a new term to be loaded into the Text Editor, so you could lose any pending changes to the cross product definition if one of these buttons is clicked (though the Text Editor should warn you that you have unsaved changes, and give you the opportunity to hit Cancel).
To add a new genus term, type part of a term name or ID into the Intersection Genus box. After a brief pause, a dropdown menu of term names that contain the characters you typed will appear; click on one to select it. (Note that the dropdown menu includes only the top ten matches to the string you entered; there may be more matching terms than that.)
There is currently no way to delete an unwanted genus in the Cross Product Editor (a genus trash can is in the works). For now, you may need to use the Parent Editor to delete unwanted genuses.
Note that it is possible to create a valid cross-product with differentia and no genus term in OBO-Edit; however, this is not recommended.
You can assign differentia terms by clicking the button that says "Click (or drop) here to add a new differentia" and then typing or pasting a term name or ID into the text box. As described above, as you start typing, an autocomplete dropdown menu will appear, which you can choose from. To assign the relationship for the differentia, use the pulldown menu of relationships.
Another way to add a differentia is to drag a term onto the "Click (or drop) here to add a new differentia" button. The easiest way to do this is to have two (or more) Ontology Tree Editor Panels open, because you will need to select the differentia term(s) in an Ontology Tree Editor panel that's in local selection mode. See the Ontology Tree Editor Panel page for more information on using multiple Ontology Tree Editor panels.
In an Ontology Tree Editor panel that is in local selection mode, select a term that you wish to use as a differentia term. Note that you cannot use the primary selection panel to select the term you wish to use as the differentia term because selecting another term will unselect the term of interest for which you are trying to make the cross product relationships.
Drop the new differentia term onto the button labelled "Click (or drop) here to add a new differentia". A popup menu will appear containing a list of all available relationship types in the current ontology. Choose an option from the menu to assign a type to your new relationship.
To remove a differentia term, click the garbage can button to the right of the unwanted differentia. If you've accidentally gotten into a state where you have a differentia but no genus, that may not work--you may need to use the Parent Editor to get rid of the differentia.
If you are in autocommit mode, changes to the cross products are automatically saved when you move on to another term. If you aren't, then click the Commit button in the Text Editor to save your changes. OBO-Edit will create the hidden links that record your new cross product. If the reasoner is running, OBO-Edit will try to create inferred is_a links based on your new cross-product definition.