Is there a notion of a smallest category satisfying some requirements? Let $\text{Cat}$ be some category (e.g. category of algebras over $\mathbb{R}$). In many typical constructions, if we want to find the smallest object $A \in \text{Obj}(\text{Cat})$  that satisfies some properties $R$, we may explicitly construct
$$A = \bigcap_{B \in \mathcal{A}}B ,$$
where $\mathcal{A} = \{B \in \text{Obj}(\text{Cat}) \: | \: B \text{ satisfies } R \}$.
Is something analogous possible for categories themselves, i.e. is there a notion of "smallest" category $C$ that satisfies conditions $R$?
 A: Your question is not very precise, but I think this will cover it. Let $\mathcal{C}$ be any category and let $\mathbf{A}$ be any (non-empty) collection of objects and arrows in $\mathcal{C}$. So $\mathbf{A}$ is itself not necessarily a category. We then define
$$
\mathcal{A} = \bigcap \{\mathcal{B} : \mathcal{B} \text{ is a subcategory of } \mathcal{C} \text{ and contains } \mathbf{A} \}.
$$
Then $\mathcal{A}$ will be the smallest subcategory of $\mathcal{C}$ containing $\mathbf{A}$.
For this we have to check that $\mathcal{A}$ is indeed a category. Throughout, $\mathcal{B}$ will denote an arbitrary subcategory of $\mathcal{C}$ containing $\mathbf{A}$.


*

*Every object $X$ in $\mathcal{A}$ has an identity arrow, because if $X$ is in $\mathcal{B}$ then $Id_X$ is in $\mathcal{B}$, hence $Id_X$ is in $\mathcal{A}$.

*If $f: X \to Y$ and $g: Y \to Z$ are in $\mathcal{A}$, then their composition $gf$ is in $\mathcal{A}$. Again this is using that $\mathcal{B}$ is a category: $f$ and $g$ are in $\mathcal{A}$, so they are in $\mathcal{B}$ hence $gf$ is in $\mathcal{B}$.

*Composition is associative and respects the identity arrows because the operation is inherited from $\mathcal{C}$.

