In Proposition 3.11.1 of Volume 2 of Borceux's Handbook of Categorical Algebra, he proves that the category of Lawvere theories is cocomplete, and his proof of this result uses the fact that every Lawvere theory has a syntactic presentation and every syntactic presentation of an algebraic theory has an associated Lawvere theory. Is there a more purely categorical (i.e. syntax-free) way to prove that the category of Lawvere theories is cocomplete?
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The category $\mathbf{Law}$ of Lawvere theories is algebraic, as described in this answer. Every algebraic category is cocomplete (in fact, locally (strongly) finitely presentable). This follows, for instance, from the theory of sketches. A good reference is Adámek–Rosický's On sifted colimits and generalized varieties.
