Based on the discussion here I feel like there should be a bijection between pseudonatural transformations of pseudofunctors $J\to\mathcal{C}$ and pseudofunctors $J\times [1]\to\mathcal{C}$, at least morally ($[1]$ denotes the poset 0<1).

The map from 'homotopies' $J\times [1]\to\mathcal{C}$ to pseudonatural transformations works out nicely, but the other direction seems problematic. In particular, given $\alpha:F\Rightarrow G$, if we try to form a pseudofunctor $\tilde{\alpha}:J\times [1]\to\mathcal{C}$ it's clear that we should set $$ \tilde{\alpha}(\text{id}_j,0\to 1) = \alpha (j),\; \text{and}\quad \tilde{\alpha}(j'\to j,\text{id}_0)=F(j'\to j),\tilde{\alpha}(j'\to j,\text{id}_1) = G(g) $$ but what about $\tilde{\alpha}(j'\to j,0\to 1)$? The problem is pseudonatural transformations only tell how to fill squares, not the triangles individually (i.p. they don't give any diagonal 1-morphism in the middle). One could just choose either $\alpha(j)\circ F(g)$ or $G(g) \circ \alpha(j)$ but this won't result in a bijection.

Is the best one can hope for an equivalence modulo modification?