I think the generalization I want is something like:

A "topological multicategory" is:
a set of objects X
for each x ∈ X, a set Hx of morphisms whose codomain is x.
for each x ∈ X, a homset operator [—, x] : 2X → 2Hx. Given a set Z of domain objects, it yields [Z, x], the set of morphisms that are allowed to go from Z to x. We write f : Z → x for f ∈ [Z, x].
for each x ∈ X, there's an identity arrow idx : {x} → x
If f : W → x and for each w ∈ W we have a gw : Z → w, then we can form the composite f ⚬ (gw)w ∈ W : Z → x.
If Z ⊆ W and f : Z → x, then also f : W → x

Presumably composition has to be associative and respect identities. I think also if V ⊆ W and f : V → x, and for all w ∈ W we have gw : Z → w, then we should have that
f ⚬ (gw)w ∈ W = f ⚬ (gv)v ∈ V
That is, if f does know how to have a more tidied up reduced domain, then composites of arrows along the irrelevant input ports shouldn't matter.

A functor from X to Y is a map F from objects of X to objects of Y, and a map of homsets
[F*Z, x] → [Z, Fx]
for any Z ⊆ Y and x ∈ X. Presumably it respects composition and identities in some appropriate way.
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Dinner with akiva and dannel at nuevo portal in carroll gardens. Ate a pile of chicken stew and rice and beans and maduros, good times. I do miss…

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