A category is $\mathbf{J}$-complete? Let $\mathbf{C}$ be a category and $\mathbf{J}$ be an index category. What does it mean to say that $\mathbf{C}$ is $\mathbf{J}$-complete? Is it just saying that all $\mathbf{J}$ shaped diagrams in $\mathbf{C}$ have a limit? 

Yes, this is exactly what it means. A category is $\mathbf{J}$-complete when it has all $\mathbf{J}$-shaped limits.
