In the paper
https://homepages.inf.ed.ac.uk/cheunen/publications/2015/matroids/matroids.pdf
the category of matroids and strong maps is defined and investigated. In Section 8, the authors take into account various matroidal constructions, following some relevant constructions by Brylawksi. Among these, the authors analyze free extensions (see Definition 8.5). In the successive remark, they show that free extensions are not functorial. Let us see the example: the ground set of the matroid $M$ is the $2$-set $\{a,b\}$ while that of the matroid $N$ is the $4$-set $\{a,b,c,d\}$. When freely extending $M$, the authors should use $p$ instead of $d$. Something on the flats of $X(N)$ should be corrected. In fact, we must have
$$ \mathcal{F}_{X(N)}=\{\emptyset,a,b,c,d,p,ab,ac,ad,bc,bd,cd,ap,bp,cp,dp,abp,acp,adp,bcp,bdp,cdp,abcdp\} $$
instead of the family given by the authors. However, it is true that there is no strong map from $X(M)$ to $X(N)$ whose restriction to $M$ and $N$ agrees with $f$. The problem is the next sentence: "Hence X(f) cannot be canonically defined in a way that respects identities, and the free extension cannot be functorial". What do the authors exactly mean?
The “that agree with $f$” part of the note shows that this extension construction cannot be made functorial in a manner allowing a natural transformation from the identity functor, something you would naturally want out of an extension operation.