Let $G$ and $H$ be Lie groupoids. We know that there are two notions of equivalences of Lie groupoids:

**Strongly equivalent Lie groupoids:**(*My terminology*)

A homomorphism $\phi:G \rightarrow H$ of Lie groupoids is called a strong equivalence if there is a Lie groupoid homomorphism $\psi:H \rightarrow G$ and natural transformation of Lie groupoid homomorphism $T: \phi \circ \psi \Rightarrow \mathrm{id}_H$ and $S: \psi \circ \phi \rightarrow \mathrm{id}_G$. *In this case $G$ and $H$ is said to be strongly equivalent Lie groupoids.*

**Weakly Equivalent or Morita Equivalent Lie groupoids**:

A homomorphism $\phi:G \rightarrow H$ of Lie groupoids is called a weak equivalence if it satisfies the *following two conditions*

where $H_0$, $H_1$ are object set and morphism set of Lie groupoid H respectively. Similar meaning holds for symbols $G_0$ and $G_1$. Here symbols $s$ and $t$ are source and target maps respectively. The notation $pr_1$ is the projection to the first factor from the fibre product. from t. Here the condition **(ES**) says about essential surjectivity and the condition **(FF)** says about full faithfulness.

*One says that two Lie Groupoids $G$ and $H$ are weakly equivalent or Morita equivalent if there exist weak equivalences $\phi:P \rightarrow G$ and $\phi':P \rightarrow H$ for a third Lie groupoid $P$.*

(According to https://ncatlab.org/nlab/show/Lie+groupoid#2CatOfGrpds one motivation for introducing Morita equivalence is the failure of the axiom of choice in the category of smooth manifolds )

**What I am looking for:**

**Now let we replace $G$ and $H$ by categories $G'$ and $H'$ which are categories internal to a category of generalized smooth spaces** (For example, category of Chen spaces or category of diffeological spaces... etc). For instance, our categories $G'$ , $H'$ can be path groupoids.

Analogous to the case of Lie groupoids I can easily define the notion of **Strongly equivalent categories internal to a category of generalized smooth spaces.**

Now if I assume that the *axiom of choice* fails also in the category of generalized smooth spaces then it seems reasonable to introduce a notion of weakly equivalent or some sort of Morita equivalent categories internal to a category of generalized smooth spaces.

But it seems that we cannot directly define the notion of **weakly equivalent or Morita equivalent categories internal to a category of Generalized Smooth Spaces** in an analogous way as we have done for Lie Groupoids. *Precisely in the condition of essential surjectivity (ES) we need a notion of surjective submersion but I don't know the analogue of surjective submersion for generalized smooth spaces*

I heard that Morita equivalence of Lie groupoids are actually something called "*Anaequivalences*" between Lie groupoids.(T*hough I don't have much idea about anafunctors and anaequivalences*).

*So my guess is that the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces has something to do with anaequivalence between categories internal to a category of generalized smooth spaces.* **Is it correct?**

**My Question is the following:**

*What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?*

**EDIT:**

*In the comments section after the answer by David Roberts we also had a discussion on the following two questions:*

Let $F: G \rightarrow H$ be a Lie groupoid Homomorphism such that $F$ is fully faithful and essentially surjective as a functor between the underlying categories. Let us also assume the $G$ and $H$ are not Morita Equivalent. Then what are the properties that Lie groupoids $G$ and $H$ has in common apart from the trivial fact that they have equivalent underlying categories?

In papers on Higher gauge theory like

*Principal 2 bundles and their Gauge 2 groups*by Christoph Wockel https://arxiv.org/abs/0803.3692 and the paper*Higher Gauge theory: 2-connections*by Baez and Schreiber https://arxiv.org/abs/hep-th/0412325**why strong equivalence is preferred over weak equivalence in the notion of Local triviality for Principal-2 bundles over a manifold?**(Here equivalence means equivalence between categories internal to a category of generalized smooth spaces)

My deep apology for asking two sufficiently different (from the original) question in the comments section.

Thank you.