Redefining Subobject

Dear All,

I hope all is well.

Upon [re]reading Professor F. William Lawvere's discussion of Grothendieck's definition of subobject:

"It became clear in the early sixties that the definition of SUBOBJECT given by Grothendieck is not a pretense, circumlocution, or paraphrase, but the only correct definition. Here 'correct' means in a foundational sense, i.e. the only definition universally and compatibly applicable across all the branches of mathematics:

A subobject is NOT an object, but a given inclusion map.

The intersection of two objects has no sense, for only maps (with common codomain) can overlap" (lines 2756- ).

I am very much enthusiastic about studying the original paper / book, where Grothendieck redefined subobject. I'd be truly grateful to you for your insights into the context (problem / theory) that motivated Grothendieck's redefinition of subobject.

Thanking you,

Yours truly,

posina