When we define internal categories of some category $\mathcal{C}$ as monads in $\mathrm{Span}(\mathcal{C})$, there is a subtlety concerning the definition of internal functors. It is not the case that the morphisms of monads (namely colax monad morphisms) give us internal functors directly: we need to ask that the 1-cell of the morphism is a left adjoint in $\mathrm{Span}(\mathcal{C})$ (which corresponds to asking that the left leg of the span be the identity, or an isomorphism). (This is related to the behaviour of profunctors in that they correspond to functors (via Yoneda) exactly when they are left adjoints.) If in considering morphisms of monads in $Span(\mathcal{C})$ we don’t require this left-adjoint condition then we end up constructing Mealy morphisms between internal categories.
‘Mealy morphisms’ are named after Mealy machines, which, in turn, were named after George H. Mealy.
Robert Paré, Mealy Morphisms of Enriched Categories, Applied Categorical Structures 20 (2012) pp 251–273, doi:10.1007/s10485-010-9238-8.
Bryce Clarke, Internal lenses as monad morphisms
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