Kerodon

Warning 1.5.1.4. To define a functor $F$ from an ordinary category $\operatorname{\mathcal{C}}$ to an ordinary category $\operatorname{\mathcal{D}}$, it suffices to specify the values of $F$ on objects and morphisms (as described in $(a)$ and $(b)$ of Remark 1.5.1.3) and to verify that $F$ is compatible with the formation of composition and identity morphisms (as described in $(c)$ and $(d)$ of Remark 1.5.1.3). In the $\infty $-categorical setting, this is not enough: to give a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, one must specify its values on simplices of all dimensions. Roughly speaking, these values encode the requirement that $F$ is compatible with composition “up to coherent homotopy.” For example, suppose that we are given objects $X,Y,Z \in \operatorname{\mathcal{C}}$ and morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$. Part $(d)$ of Remark 1.5.1.3 asserts that if $h$ is a composition of $f$ and $g$, then $F(h)$ is a composition of $F(f)$ and $F(g)$. However, we can say more: if $\sigma $ is a $2$-simplex of $\operatorname{\mathcal{C}}$ which witnesses $h$ as a composition of $f$ and $g$, then $F(\sigma )$ is a $2$-simplex of $\operatorname{\mathcal{D}}$ which witnesses $F(h)$ as a composition of $F(f)$ and $F(g)$.