# An embarrassing question about subobjects

In Robert Goldblatt "Topoi, The Categorial Analysis Of Logic", in the beginning of Ch4 about subobjects, it reads "Now if $$f \subseteq g$$ and $$g \subseteq f$$ then $$f$$ and $$g$$ each factor through each other, as in [diagram...] $$f = g \circ h$$ and $$g = f \circ i$$. In that case, $$h \colon a \rightarrow b$$ is iso".

My attempt at showing this starts by seeing that $$g = g \circ (h \circ i)$$ and $$f = f \circ (i \circ h)$$ - but how do I conclude that $$h \circ i = 1_a$$ (where $$a = dom(f)$$). It feels like it ought to be a very simple, algebraic step; it just eludes me at the moment.

• Recall the definition of monomorphism! Jan 20, 2020 at 16:24
• hmm, ... how to typeset 'facepalm' in LaTeX.... Well, I did say it ought to be simple and I ought to be embarrassed. Thank you for proving me right :-) Jan 20, 2020 at 16:31