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Example 2.1.5.21 (Algebra Objects). Let $\operatorname{\mathcal{C}}$ be a monoidal category. An algebra object of $\operatorname{\mathcal{C}}$ is a pair $(A,m)$, where $A$ is an object of $\operatorname{\mathcal{C}}$ and $m: A \otimes A \rightarrow A$ is a nonunital algebra structure on $A$ (Example 2.1.4.11) for which there exists a unit $\epsilon : \mathbf{1} \rightarrow A$ (in the sense of Definition 2.1.5.1). If $(A,m)$ and $(A',m')$ are algebra objects of $\operatorname{\mathcal{C}}$ with units $\epsilon : \mathbf{1} \rightarrow A$ and $\epsilon ': \mathbf{1} \rightarrow A'$, then we say that a morphism $f: A \rightarrow A'$ is an algebra homomorphism if it is a nonunital algebra homomorphism (Example 2.1.4.11) which satisfies $\epsilon ' = f \circ \epsilon$. We let $\operatorname{Alg}(\operatorname{\mathcal{C}})$ denote the category whose objects are algebra objects of $\operatorname{\mathcal{C}}$ and whose morphisms are algebra homomorphisms. We regard $\operatorname{Alg}(\operatorname{\mathcal{C}})$ as a (non-full) subcategory of the category $\operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$ of nonunital algebra objects of $\operatorname{\mathcal{C}}$ defined in Example 2.1.4.11.

Let $\{ e\}$ denote the trivial monoid, regarded as a (strict) monoidal category having only identity morphisms (Example 2.1.1.3). Then algebra objects of $\operatorname{\mathcal{C}}$ can be identified with lax monoidal functors $\{ e\} \rightarrow \operatorname{\mathcal{C}}$. More precisely, the isomorphism $\operatorname{Fun}^{\operatorname{lax}}_{\operatorname{nu}}( \{ e \} , \operatorname{\mathcal{C}}) \simeq \operatorname{Alg}^{\operatorname{nu}}(\operatorname{\mathcal{C}})$ of Example 2.1.4.11 specializes to an isomorphism of (non-full) subcategories $\operatorname{Fun}^{\operatorname{lax}}( \{ e\} , \operatorname{\mathcal{C}}) \simeq \operatorname{Alg}(\operatorname{\mathcal{C}})$.