# Kerodon

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Remark 2.2.4.19. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a unitary lax functor. Then one can modify $F$ to produce a strictly unitary lax functor $F': \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ by the following explicit procedure:

• For every object $X \in \operatorname{\mathcal{C}}$, we set $F'(X) = F(X)$.

• For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ which is not an identity morphism, we set $F'(f) = F(f)$; if $X = Y$ and $f = \operatorname{id}_{X}$ we instead set $F'(f) = \operatorname{id}_{F(X)}$. In either case, we have an invertible $2$-morphism $\varphi _{f}: F'(f) \xRightarrow {\sim } F(f)$, given by

$\varphi _{f} = \begin{cases} \epsilon _{X}^{F} & \text{ if } f = \operatorname{id}_ X \\ \operatorname{id}_{ F(f) } & \text{ otherwise. } \end{cases}$
• Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, and let $\gamma : f \Rightarrow g$ be a $2$-morphism between $1$-morphisms $f,g: X \rightarrow Y$. We define $F'(\gamma )$ to be the vertical composition $\varphi _{g}^{-1} F(\gamma ) \varphi _{f}$.

• For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in the $2$-category $\operatorname{\mathcal{D}}$, we define the composition constraint $\mu ^{F'}_{g,f}: F'(g) \circ F'(f) \Rightarrow F'(g \circ f)$ to be the vertical composition

$F'(g) \circ F'(f) \xRightarrow { \varphi _{g} \circ \varphi _{f} } F(g) \circ F(f) \xRightarrow { \mu _{g,f}^{F}} F(g \circ f) \xRightarrow { \varphi _{g \circ f}^{-1} } F'( g \circ f ).$

Consequently, it is generally harmless to assume that a unitary lax functor of $2$-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is strictly unitary.