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.

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    $\begingroup$ Recall the definition of monomorphism! $\endgroup$ Jan 20, 2020 at 16:24
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    $\begingroup$ 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 :-) $\endgroup$
    – j4nd3r53n
    Jan 20, 2020 at 16:31


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