9.7.4 Pseudomonic Functors
Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
A functor $F\colon \mathcal{C}\to \mathcal{D}$ is pseudomonic if it satisfies the following conditions:
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For all diagrams of the form
if we have
\[ \text{id}_{F}\mathbin {\star }\alpha =\text{id}_{F}\mathbin {\star }\beta , \]
then $\alpha =\beta $.
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For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$ and each natural isomorphism there exists a natural isomorphism such that we have an equality of pasting diagrams, i.e. such that we have
\[ \beta =\text{id}_{F}\mathbin {\star }\alpha . \]
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
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Characterisations. The following conditions are equivalent:
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The functor $F$ is pseudomonic.
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The functor $F$ satisfies the following conditions:
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The functor $F$ is faithful, i.e. for each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the action on morphisms
\[ F_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{D}}\webleft (F_{A},F_{B}\webright ) \]
of $F$ at $\webleft (A,B\webright )$ is injective.
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For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the restriction
\[ F^{\textup{iso}}_{A,B} \colon \textup{Iso}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Iso}_{\mathcal{D}}\webleft (F_{A},F_{B}\webright ) \]
of the action on morphisms of $F$ at $\webleft (A,B\webright )$ to isomorphisms is surjective.
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We have an isocomma square of the form in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
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We have an isocomma square of the form in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
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For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition1 functor
\[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]
is pseudomonic.
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Conservativity. If $F$ is pseudomonic, then $F$ is conservative.
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Essential Injectivity. If $F$ is pseudomonic, then $F$ is essentially injective.
Omitted.
Omitted.
Omitted.
