In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors

Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors.

Let ${\displaystyle ({\mathcal {C}},\otimes ,I_{\mathcal {C}})}$ and ${\displaystyle ({\mathcal {D}},\bullet ,I_{\mathcal {D}})}$ be monoidal categories. A lax monoidal functor from ${\displaystyle {\mathcal {C}}}$ to ${\displaystyle {\mathcal {D}}}$ (which may also just be called a monoidal functor) consists of a functor ${\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}$ together with a natural transformation

between functors ${\displaystyle {\mathcal {C}}\times {\mathcal {C}}\to {\mathcal {D}}}$ and a morphism

called the coherence maps or structure morphisms, which are such that for every three objects ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$ of ${\displaystyle {\mathcal {C}}}$ the diagrams

commute in the category ${\displaystyle {\mathcal {D}}}$. Above, the various natural transformations denoted using ${\displaystyle \alpha ,\rho ,\lambda }$ are parts of the monoidal structure on ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {D}}}$.

If ${\displaystyle ({\mathcal {C}},\otimes ,I_{\mathcal {C}})}$ and ${\displaystyle ({\mathcal {D}},\bullet ,I_{\mathcal {D}})}$ are closed monoidal categories with internal hom-functors ${\displaystyle \Rightarrow _{\mathcal {C}},\Rightarrow _{\mathcal {D}}}$ (we drop the subscripts for readability), there is an alternative formulation

of φ_AB commonly used in functional programming. The relation between ψ_AB and φ_AB is illustrated in the following commutative diagrams: