Smash product of categories

I'm playing with some homotopy stuff and I want to figure out what a smash product over categories ought to be.

Confusingly pointed categories means something else but I just want something like pointed sets, a category and an object.

Now I feel there are three reasonable interpretations of a "one-object" functor $ F : (C, c) \rightarrow (D, d) $.

1. A "strict one-object functor", mapping point to point $\{ F : C \rightarrow D \mid F(c) = d \}$
2. A "nonstrict one-object functor" mapping point to isomorphic points $\Sigma \, F : C \rightarrow D , \, F(c) \leftrightarrow d $
3. A "lax one-object functor" mapping a point to a coercible point $\Sigma \, F : C \rightarrow D , \, d \rightarrow F(c)$

Anyhow you now want a product $\wedge$ adjoint to the internal hom. I feel like there are a few different interpretations of this and I'm confused between the strict, nonstrict and lax versions.

You obviously want to base the smash product around the product category but I find it unclear how to think about quotienting the points.

$$ (C, c) \wedge (D, d) = (C \times D/(c, y)\sim(x, d), (c, d)) $$

In the nonstrict versions you don't need any different objects than the product category. You just have to add in an isomorphism $(c,y) \leftrightarrow (x, d)$ or a one-way map $(x, d) \rightarrow (c, y)$.

I don't really see how you'd hack the morphisms of the product category though.

See Construction 4.19 here:

This is a tensor-hom adjunction.