Is there any treatment on principal "categorical" bundles - principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is some (topological) category?

I know that one can define "categorical covering spaces" - locally constant $\mathbf{C}$-valued sheaves on a topological space $X$. I think that there's a nice equivalence of categories of the form

$$\left\{\text{Locally constant } \mathbf{C}-\text{sheaves on }X \right\} \leftrightarrow [\Pi_1(X), \mathbf{C}]$$

where $\Pi_1(X)$ is the fundamental groupoid of $X$ and $[\_,\_]$ is the functor category.

Is there a general notion of principal $\mathbf{C}-$bundle that captures this particular case? Is there any hope of a nice result in the form of a bijection

$$\left\{ \text{Principal } \mathbf{C}-\text{bundles on } X \text{ up to iso} \right\} \leftrightarrow [X, B \mathbf{C}]$$

where $[X, B\mathbf{C}]$ represents homotopy classes of maps from $X$ to the classifying space of the category $\mathbf{C}$?