Cartier and Weil divisors agree on a smooth curve

The local rings. Let $p$ be a closed point of $C$; by [F1] the local ring ${\mathcal{O}}_{C,p}$ is a discrete valuation ring, and by [F2] it is a principal ideal domain, hence a unique factorisation domain. Let $\eta$ be the generic point of the integral scheme $C$; by [F1] its local ring is the function field $k(C)$, a field, hence again a unique factorisation domain by [F2]. These are all the points of the one-dimensional space $C$, so every local ring of $C$ is a unique factorisation domain.

Divisors for invertible sheaves. Let $\mathcal{L}$ be an invertible sheaf on $C$. By [F6] the stalk of $\mathcal{L}$ at the generic point is a one-dimensional $k(C)$-vector space, so $\mathcal{L}$ admits a nonzero rational section $s$; by the current interface thm-line-bundle-rational-section-cartier-divisor of [F4] there is a Cartier divisor ${\mathrm{div}}_C(s)$ with ${\mathcal{O}}_C({\mathrm{div}}_C(s))\cong \mathcal{L}$, so every invertible sheaf is of the form ${\mathcal{O}}_C(D)$ for a divisor $D$. If ${\mathcal{O}}_C(D)\cong {\mathcal{O}}_C(D^{\mathrm{\prime }})$, then $D-D^{\mathrm{\prime }}$ lies in the kernel of $D\mapsto [{\mathcal{O}}_C(D)]$, which is ${\mathrm{Prin}}_C(C)$ by the current interface thm-cartier-divisors-mod-principal-to-picard of [F4], i.e. $D-D^{\mathrm{\prime }}$ is a principal Cartier divisor; by [F4] and [F5] this is linear equivalence, and by the definition def-linear-equivalence-cartier-divisors of [F4] the divisor $D$ is well defined modulo linear equivalence, the second half of clause 3.

Local factoriality. By step 1.1 every local ring of $C$ is a unique factorisation domain, so the defining condition of [F3] holds and $C$ is locally factorial; the curve is also Noetherian and integral by [F1], which are the structural hypotheses of the current suppliers below.

Cartier and Weil divisors agree. By the current interface thm-cartier-weil-isomorphism-locally-factorial of [F4], every Weil divisor on the locally factorial Noetherian integral scheme $C$ is locally Cartier, hence Cartier (the Cartier condition is local), and the cycle map is an isomorphism onto the Weil divisor group. The current interface thm-cartier-to-weil-divisor-normal-scheme of [F4] supplies the well-definedness of the cycle map $\mathrm{cyc}\text{\hspace{0.17em}⁣}:\mathrm{CaDiv}(C)\to \mathrm{Div}(C)$ and its compatibility $\mathrm{cyc}({\mathrm{div}}_C(f))={\mathrm{div}}_W(f)$ with principal divisors, so $\mathrm{cyc}$ identifies $\mathrm{CaDiv}(C)$ with $\mathrm{Div}(C)$ and carries linear equivalence to linear equivalence; this is clause 2 of the Statement.

The Picard group. By the current interface thm-cartier-divisors-mod-principal-to-picard of [F4] the assignment $D\mapsto [{\mathcal{O}}_C(D)]$ induces an isomorphism $\mathrm{CaDiv}(C)\mathrm{/}{\mathrm{Prin}}_C(C)\to \mathrm{Pic}(C)$, the curve $C$ being integral by [F1]; by step 3.1 the cycle map identifies the source with $\mathrm{Div}(C)\mathrm{/}\mathrm{Prin}(C)=\mathrm{Cl}(C)$, using that ${\mathrm{Prin}}_C(C)$ maps onto $\mathrm{Prin}(C)$ by the compatibility of the cycle map with principal divisors and [F5]. Composing these isomorphisms gives $\mathrm{Pic}(C)\cong \mathrm{Cl}(C)$, the first half of clause 3.

Conclusion. The local rings of $C$ at closed points are discrete valuation rings, hence principal ideal domains and unique factorisation domains, and the local ring at the generic point is a field, so $C$ is locally factorial by steps 1.1 and 2.1, which is clause 1. On the locally factorial Noetherian integral curve the current cycle and class-group interfaces of [F4] identify Cartier with Weil divisors and the Picard group with the divisor class group, by steps 3.1 and 4.1, which is clauses 2 and 3, and every invertible sheaf is ${\mathcal{O}}_C(D)$ with $D$ well defined modulo linear equivalence by step 1.2. The Axiom of Choice is used through [F2], and the Dependent Choice assumed in the Statement is used exactly through the two cycle suppliers of [F4]; no other choice principle is invoked.