Here1, on the page 282, I would like to understand why precisely Examples of $k$-ary operations are all $k-\mathrm{lim}$, BUT $k-\textrm{colim}$ must be filtered. Where have we used that $k$ is filtered and how taking $k$-colim becomes an operation? And yet, on that page 282 above those examples, what is the difference between $\ast$ and $\cdot$ in the equation $$\omega_{G\cdot F}=(G\ast\omega_F)\cdot (\omega_G\ast F)?$$ What squares these $\ast$ and $\cdot$ amounts to commute? I more or less understand what comes next in the paper, but for this page.
1 Jiří Adámek and Jiří Rosický: Algebra and local presentability: how algebraic are they? (A survey); Tbilisi Mathematical Journal 10(3) (2017), pp. 279–295; DOI: 10.1515/tmj-2017-0113
Morphisms of varieties preserve limits and filtered colimits, but not general colimits. This is essentially the definition of an operation on VAR. The $*$ denotes whiskering a functor with a natural transformation, while $\cdot$ denotes (vertical) composition of natural transformations. All the authors are doing here is spelling out the meaning of pseudonatural transformation, which you can find in more verbose form in many places.