= Solution
Let the <first cousins> be $X,Y$, with parents $(P,U)$ and $(Q,V)$ respectively, where $P,Q$ are <full siblings> and $U,V$ have no ancestral relationship to one another or to that sibling family. To compute $\phi_{XY}$, choose one <allele> copy from each cousin. They originate from $P$ and $Q$ with <probability> $1/4$. All other parental combinations have zero <kinship coefficient> under the otherwise <outbred pedigree> assumption. Hence
$$
\phi_{XY}=\frac14\phi_{PQ}=\frac14\cdot\frac14=\frac1{16}.
$$
The two gene copies within a child of $X,Y$ are random transmissions from these parents. They are <identical by descent> precisely in the event defining parental kinship, so the child's <inbreeding coefficient> is
$$
\boxed{F_{\mathrm{child}}=\phi_{XY}=\frac1{16}.}
$$
For a direct ancestral-path check, there are two shared noninbred grandparents. Through either grandparent, the two cousin-to-grandparent paths have two <meioses> each, and matching the ancestral grandparental copy supplies a further factor $1/2$. Each contributes $2^{-2-2-1}=1/32$, giving the same <first-cousin inbreeding coefficient>. Additional ancestral relationships or inbred grandparents would change the result.
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