I'm looking for an opinion if the following makes sense.

A **linear representation** of a groupoid $\mathcal{G}$ is a functor $$\nabla: \mathcal{G}\longrightarrow \mathsf{Vect}_{\mathbb K},$$ where $\mathsf{Vect}_{\mathbb K}$ is the category of vector spaces over $\mathbb K$.

I presume we could define analogously a **linear 2-representation** of a strict $2$-grupoid $2\textrm{-}\mathcal{G}$ as a strict 2-functor:

$$2\textrm{-}\nabla: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Vect}_{\mathbb K},$$ where $2\textrm{-}\mathsf{Vect}_{\mathbb K}$ is the $2$-category of 2-vector spaces (the objects are groupoids internal to $\mathsf{Vect}_{\mathbb K}$). I don't know if this definition appears in the literature but I believe it sounds reasonable.

Now, how to make sense of a non-linear version of a 2-representation of a 2-groupoid?

I'm inclined to define a **2-action** of a strict $2$-groupoid $2\textrm{-}\mathcal{G}$ is a strict two functor

$$\rho: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Grpd},$$ where $2\textrm{-}\mathsf{Grpd}$ is the 2-category of 2-grupoids (the objects are groupoids internal to $\mathsf{Grpd}$).

To me it sounds $2\textrm{-}\mathsf{Grpd}$ must be the non-linear version of $2\textrm{-}\mathsf{Vect}_{\mathbb K}$.

But I'm not quite certain if this is really coeherent with what an action must mean. I believe category theory must provide some prototype of what an action must be in any setting and I should follow that, but that is beyond my categorical knowledge.

To be precise, my question is, is it "right" to define a 2-action of a strict 2-groupoid as strict 2-functor $\rho: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Grpd}$?

I'm looking for arguments which could support it is "natural" to make such definition.

Thanks.