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After looking at some materials, I think a good candidate is ^-comp standing for comprison maps. Generally, when some functor is expected to respect some structure, there's often already a onesided map. So we can similarly have ×-comp which is roughly F(X×Y) → FX × FY etc.
See PR #339 by @Trebor-Huang for the details. The current names are sub-optimal.
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