At the end of the introduction of the article "Internal Graphs and Internal Groupoids in Mal'cev Categories", Carboni, Pedicchio and Pirovano write as follows:

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> While working in a category with finite limits, we freely use elements as far as Horn sentences built up from atomic ones and equality are concerned; we use existential quantification only when we can show uniqueness. It is well known that in such a way we can always interpret our statement in a lex category in a valid way.
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I don't understand what they are saying. In the article they use elements more or less in every definition, statement or proof, and I don't understand how is this possible since the context is the one of Mal'cev categories. What general results are they using in order to do categorical proofs by using elements?

Any help would be appreciated.