Remark 1.5.1.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
- $(a)$
To each object $X \in \operatorname{\mathcal{C}}$ the functor $F$ assigns an object of $\operatorname{\mathcal{D}}$, which we will denote by $F(X)$ (or sometimes more simply by $FX$).
- $(b)$
To each morphism $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the functor $F$ assigns a morphism $F(f): F(X) \rightarrow F(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$.
- $(c)$
For every object $X \in \operatorname{\mathcal{C}}$, the functor $F$ carries the identity morphism $\operatorname{id}_{X}: X \rightarrow X$ in $\operatorname{\mathcal{C}}$ to the identity morphism $\operatorname{id}_{F(X)}: F(X) \rightarrow F(X)$ in $\operatorname{\mathcal{D}}$.
- $(d)$
If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are morphisms in $\operatorname{\mathcal{C}}$ and $h$ is a composition of $f$ and $g$ (in the sense of Definition 1.4.4.1), then the morphism $F(h): F(X) \rightarrow F(Z)$ is a composition of $F(f)$ and $F(g)$.