Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A **functor** $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ consists of 1. An assignment on objects: for every object $X \in \mathrm{Ob}(\mathcal{C})$, an object $F(X) \in \mathrm{Ob}(\mathcal{D})$. 2. An assignment on morphisms: for every morphism $f \colon X \to Y$ in $\mathcal{C}$, a morphism $F(f) \colon F(X) \longrightarrow F(Y)$ in $\mathcal{D}$. Moreover, these assignments must satisfy: - **Preservation of identities**: For every object $X$ in $\mathcal{C}$, $F(\mathrm{id}_X) \;=\; \mathrm{id}_{F(X)}$. - **Preservation of composition**: For every pair of composable morphisms $X \xrightarrow{\,f\,} Y \xrightarrow{\,g\,} Z$ in $\mathcal{C}$, $F(g \circ f) \;=\; F(g) \,\circ\, F(f)$. --- 1. **Action on a single morphism** <!-- https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzEsMCwiWSJdLFswLDEsIkYoWCkiXSxbMSwxLCJGKFkpIl0sWzAsMSwiZiJdLFswLDIsIkYiLDFdLFsxLDMsIkYiLDFdLFsyLDMsIkYoZikiXV0= --> <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzEsMCwiWSJdLFswLDEsIkYoWCkiXSxbMSwxLCJGKFkpIl0sWzAsMSwiZiJdLFswLDIsIkYiLDFdLFsxLDMsIkYiLDFdLFsyLDMsIkYoZikiXV0=&embed" width="620" height="500" style="border-radius: 8px; border: none;"></iframe> 2. **Preservation of identity morphisms** <!-- https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzEsMCwiWCJdLFswLDEsIkYoWCkiXSxbMSwxLCJGKFgpIl0sWzAsMSwiXFxtYXRocm17aWR9X1giXSxbMCwyLCJGIiwxXSxbMSwzLCJGIiwxXSxbMiwzLCJcXG1hdGhybXtpZH1fe0YoWCl9Il1d --> <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzEsMCwiWCJdLFswLDEsIkYoWCkiXSxbMSwxLCJGKFgpIl0sWzAsMSwiXFxtYXRocm17aWR9X1giXSxbMCwyLCJGIiwxXSxbMSwzLCJGIiwxXSxbMiwzLCJcXG1hdGhybXtpZH1fe0YoWCl9Il1d&embed" width="620" height="500" style="border-radius: 8px; border: none;"></iframe> 3. **Preservation of composition** <!-- https://q.uiver.app/#q=WzAsNixbMCwwLCJYIl0sWzEsMCwiWSJdLFsyLDAsIloiXSxbMCwxLCJGKFgpIl0sWzEsMSwiRihZKSJdLFsyLDEsIkYoWikiXSxbMCwxLCJmIiwyXSxbMCwzLCJGIiwxXSxbMSwyLCJnIiwyXSxbMSw0LCJGIiwxXSxbMiw1LCJGIiwxXSxbMyw0LCJGKGYpIl0sWzQsNSwiRihnKSJdLFswLDIsImcgXFxjaXJjIGYiLDIseyJjdXJ2ZSI6LTN9XSxbMyw1LCJGKGcpIFxcY2lyYyBGKGYpIiwwLHsiY3VydmUiOjR9XV0= --> <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsNixbMCwwLCJYIl0sWzEsMCwiWSJdLFsyLDAsIloiXSxbMCwxLCJGKFgpIl0sWzEsMSwiRihZKSJdLFsyLDEsIkYoWikiXSxbMCwxLCJmIiwyXSxbMCwzLCJGIiwxXSxbMSwyLCJnIiwyXSxbMSw0LCJGIiwxXSxbMiw1LCJGIiwxXSxbMyw0LCJGKGYpIl0sWzQsNSwiRihnKSJdLFswLDIsImcgXFxjaXJjIGYiLDIseyJjdXJ2ZSI6LTN9XSxbMyw1LCJGKGcpIFxcY2lyYyBGKGYpIiwwLHsiY3VydmUiOjR9XV0=&embed" width="620" height="500" style="border-radius: 8px; border: none;"></iframe> These diagrams express that $F$ is a structure-preserving map from one category into another, capturing all objects and morphisms (arrows) in a manner consistent with composition and identities.