= Solution
{c}
A <reflexive pair> $f,g:A\rightrightarrows B$ has a common section $r:B\to A$ with $fr=gr=1_B$. Form the <pushout in a category> of $f$ and $g$, and write its two maps from $B$ as $j_1,j_2$. Its relation $j_1f=j_2g$, composed with $r$, gives $j_1=j_2=q$. Thus $qf=qg$. Any $h:B\to C$ with $hf=hg$ gives the compatible pair $(h,h)$ for this pushout and therefore factors uniquely through $q$. This proves the <coequalizer of a reflexive pair from a pushout> construction.
There is a genuine transcription difference: the original PDF says finite products in the finite-colimit assertion, while the TeX says finite coproducts. The PDF assertion is false with products. Even retaining a pushout-existence assumption, the full subcategory of the <Category of sets> on nonempty sets has finite products, pushouts, and all coequalizers, but no <initial object>, hence no empty colimit. Regard the poset with elements $0,a,b,u_0,u_1,\ldots$, order
$$
0<a,b<u_{n+1}<u_n\quad(n\geq0),
$$
and incomparable $a,b$ as a <category>. It has finite meets and top $u_0$, hence finite <products in a category>. Every <reflexive pair> in a poset is an equal pair and has its identity as a <coequalizer>. But $a,b$ have no least upper bound, so they have no <coproduct in a category>.
For the corrected construction using finite coproducts, let $D:\mathcal J\to\mathcal C$ be a finite <diagram in a category>. Put
$$
X=\coprod_{j\in\mathcal J}D(j),\qquad
Y=\coprod_{a:i\to j}D(i).
$$
Define $s,t:Y\rightrightarrows X$ on the summand indexed by $a:i\to j$ as $\nu_i$ and $\nu_jD(a)$. Maps $h:X\to Z$ with $hs=ht$ are exactly <cocone under a diagram> data on $D$. Hence their universal <coequalizer> is its <colimit>. Replace $s,t$ by the <reflexive pair>
$$
[s,1_X],[t,1_X]:Y\amalg X\rightrightarrows X,
$$
whose common section is the second injection and whose coequalizer is unchanged. This proves <construction of finite colimits from coproducts and reflexive coequalizers>, including the empty diagram via the empty coproduct:
$$
\boxed{\text{Finite coproducts and reflexive coequalizers suffice for all finite colimits.}}
$$
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