Remark 1.5.4. Note that this means that two functors \(G,H\colon T \to C\) are naturally isomorphic whenever the functors \(F\circ G\) and \(F \circ H\) are naturally isomorphic, reflecting the usual categorical notion of monomorphism.
We may sometimes write \(F\colon C \hookrightarrow D\) if \(F\) is a monomorphism. If \(y\) is an object of \(D\), we say that \(y\) is contained in \(C\), sometimes written as \(y \in C\), if there exists some object \(x\) of \(C\) satisfying \(F(x) \cong y\).
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