In "Functors on locally finitely presented additive categories", by H. Krause, one can read (in page 108):

Proposition 2.3. There is, up to equivalence, a bijective correspondence between (skeletally small) additive categories with split idempotents and pseudo-kernels and (skeletally small) abelian categories with sufficiently many projectives. The correspondence is given by: $$\mathsf{C \longmapsto mod~C} ~~~~~ and ~~~~~ \mathcal{A} \longmapsto \mathsf{proj}(\mathcal{A})$$

in proof, the author says:

If $\mathsf{C}$ is an additive category with split idempotents and pseudokernels, then $\mathsf{mod~C}$ is abelian and the representable functors $\mathsf{Hom(−,X)}$ are precisely the projective objects in $\mathsf{mod~C}$ by Yoneda’s lemma. Conversely, given an abelian category A with sufficiently many projectives, $\mathsf{proj}(\mathcal{A})$ has pseudo-kernels and the inclusion $\mathsf{proj}(\mathcal{A}) \to \mathcal{A}$ extends to a functor $\mathsf{mod(proj}(\mathcal{A})) \to \mathcal{A}$ which is an equivalence.

I tried very much and studied different books and articles about this subject but I cannot understand how I can extend a functor (the following part in proof):

... the inclusion $\mathsf{proj}(\mathcal{A}) \to \mathcal{A}$ extends to a functor $\mathsf{mod(proj}(\mathcal{A})) \to \mathcal{A}$ which is an equivalence.

and also, I cannot understand why the new functor is an equivalence. there are lots of examples like this that I cannot understand. For instance, in my past question in the forum, "equivalences induced by functors", it is not clear for me how $\mathsf{mod~C} \longrightarrow \mathsf{Mod~C}$ extends to $\mathsf{Ind~mod~C \overset{\sim} \longrightarrow Mod~C}$ and why this is an equivalence. I really need some help to understand these kind of proofs by your comments or recommending some good books or articles.

$\mathcal{A}$ : An abelian category
$\mathsf{C}$ : An additive category
$\mathsf{proj}(\mathcal{A})$ : the full subcategory of $\mathcal{A}$ consisting of projective objects
$\mathsf{Mod~C}$ : The category of additive functors $\mathsf{F:C^{OP}⟶Ab}$
$\mathsf{mod~C}$ : The full subcategory of $\mathsf{Mod~C}$ consisting of finitely presented objects
$\mathsf{Ind~mod~C}$ : The full subcategory of $\mathsf{((mod~C)^{OP},Ab)}$ consisting of functors which can be expressed as filtered colimits of representable ones.

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    $\begingroup$ For existence the author uses the previously stated "Universal property 2.1". Another good reference to read about this correspondence is [Auslander: Coherent functors, 1966]. $\endgroup$ – Julian Kuelshammer Jul 18 '18 at 11:06
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    $\begingroup$ And it's an equivalence because having sufficiently many projectives here means exactly that everything is a cokernel of a map of projectives, and such cokernels are exactly what one adjoins to A in constructing A-mod. It seems you could remove the pseudo-kernels assumption on one side and the abelian assumption on the other to get a cleaner and more general statement. $\endgroup$ – Kevin Carlson Jul 18 '18 at 17:25
  • $\begingroup$ @KevinCarlson If we remove abelian assumption, proj(A) is not additive anymore. right? $\endgroup$ – math16 Jul 19 '18 at 4:41
  • $\begingroup$ @math16 I don't see why not. It just won't have pseudo-kernels, as Krause says. $\endgroup$ – Kevin Carlson Jul 19 '18 at 6:34
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    $\begingroup$ @KevinCarlson What is the additive structure on proj(A)? Shouldn't it be inherited from A? $\endgroup$ – math16 Jul 19 '18 at 8:05

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