Charge conjugation matrix in baryon current

In his paper Calculation of baryon masses in quantum chromodynamics (ScienceDirect), B.L. Ioffe considers currents describing baryons. In equation (13) he gives an interpolating current for the isobar $\Delta^{++}$, $$\eta_\mu (x) = \left( u^a(x) C \gamma_\mu u^b(x) \right) u^c(x) \varepsilon^{abc},$$ where $u^a(x)$ is an up quark field of colour $a$ and $C$ is the "charge conjugation matrix". This current has the proper isospin of $3/2$ (three up quarks) and has the correct spin (carries one Lorentz index and implicitly one spinor index at $u^c$).

How does $C$ come into this equation though? If I understand the expression correctly, $u^a(x)$ is implicitly transposed so that the expression in the big parentheses is just a number in spinor space. So I do not think that $u^a(x) C$ is some other way of writing $\bar u^a(x)$ – in which case the current would consist of two quarks and one antiquark. What is $C$ doing then?

Furthermore, how is this $C$ related to the charge conjugation matrix (say $\tilde C$ to distinguish) introduced e.g. in Peskin/Schroeder? $\tilde C$ exchanges particles with antiparticles by $\tilde C a_{\mathbf p}^s \tilde C = b_{\mathbf p}^s$ and $\tilde C b_{\mathbf p}^s \tilde C = a_{\mathbf p}^s$ resulting in e.g. $$\tilde C \psi \tilde C = - i \left(\bar\psi \gamma^0\gamma^2\right)^T$$ and seems to always appear in pairs.