There are left adjoints which are also difficult to calculate. The colimit of a small diagram of globular $\omega$-categories is a left adjoint. It is difficult to calculate because the colimit contains all free compositions of all maps of all dimensions, divided by the relations they are supposed to satisfy in a globular $\omega$-category.

What is a laborious completion ? Something difficult to calculate ?

The constructions in everyday's combinatorial model categories (e.g. the cofibrant or fibrant replacement functors) are "small" in the sense that they depend on a transfinite cardinal like $\aleph_0$, $\aleph_1$ or $\aleph_2$. The cardinals $\aleph_0$, $\aleph_1$ or $\aleph_2$ do not change with a change of universe.

There is a good reference about concrete categories: katmat.math.uni-bremen.de/acc/acc.pdf (if you don't know it already).

@KevinCarlson Yes this proves that the topological spaces are the models of a large (i.e. non-small) limit sketch. My answer was about your very last question. You can find an explicit description of the axioms in Example 5.4 of J. Rosický's paper (which is in open access, you just have to google the title).

@მამუკაჯიბლაძე It's Gabriel-Ulmer duality. It is described on both directions in the paper numdam.org/article/CTGDC_2001__42_1_51_0.pdf page 68.

@JiříRosický It may be useful to point out that an erratum for your book is available here: tu-braunschweig.de/Medien-DB/iti/cor.pdf

What's wrong with "the sets of ZFC" (what a weird expression !), and with topological spaces ? Topological spaces are precisely designed to study all these notions of connectedness, continuity, etc... It is not the starting point, but the conclusion of a very long historical process to formalize real analysis. What you need is a good book about the history of mathematics. The notion of topological space has various generalizations, Grothendieck topologies, locales more adapted to other parts of mathematics.

@PeterLeFanuLumsdaine I mean strong orthogonality, otherwise I use the word injectivity.

I perfectly understand your point. The motivation of the question is that in the particular situation I am working on, it is true. And I cannot figure out what $G$ has special. To prove it, I have to make boring annoying (but easy) calculations. I was just wondering whether I was not missing a categorical argument. In my situation, the canonical diagram with respect to $G$ is not even filtered. Maybe there is no way for me to avoid to make the calculations.