Preservation of $\diamondsuit$ by ccc forcings of size $\leq \omega_1$ This is essentially exercise H8 (p.248) of Kunen's Set Theory: An Introduction to Independence Proofs (old edition), or exercise IV.7.58 (p.307) of Kunen's Set Theory (new edition).

Suppose $P$ is a notion of forcing in $M$ such that $\left | P \right | \leq \omega_{1}$ and $P$ is ccc. Suppose further $\Diamond$ holds in $M$. How does one show that $\Diamond$ also holds $M[G]$?

Here 


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*$M$ is a countable transitive model (of $\mathsf{ZFC}$), and $M[G]$ is a generic extension of $M$ by the forcing $P$.

*$P$ being ccc (countable chain condition) means that all antichains (sets of pairwise incompatible conditions) in $P$ are countable.

*$\diamondsuit$ is the usual diamond principle:

There is sequence $\langle A_\alpha : \alpha < \omega_1 \rangle$ such that $A_\alpha \subseteq \alpha$ for each $\alpha < \omega_1$, and for each $A \subseteq \omega_1$ the set $$\{ \alpha < \omega_1 : A \cap \alpha = A_\alpha \}$$ is stationary.

 A: Let $\vec A=\langle A_\alpha\mid\alpha<\omega_1\rangle$ witness $\Diamond$ in $M$. Since $|P|\leq\omega_1$, we might as well assume that the underlying set of $P$ is a subset of $\omega_1$. By standard coding techniques, we may view a subset of $\omega_1$ as coding a $P$-name for a subset of $\omega_1$. So in $M[G]$, define $B_\alpha=\dot (A_\alpha)_G\cap\alpha$, if $A_\alpha$ codes the $P$-name $\dot A_\alpha$. 
I claim that $\langle B_\alpha\mid\alpha<\omega_1\rangle$ is a $\Diamond$ sequence in $M[G]$. To see this, fix any $A\subset \omega_1$ in $M[G]$. So there is a $P$-name $\dot A$ such that $A=\dot A_G$. Let $A^*\subset\omega_1$ be a code of $\dot A$. By a closure argument, using the fact that $P$ is c.c.c., there is a club $C\subset\omega_1$ such that $A^*\cap\alpha$ codes $\dot A\upharpoonright \alpha$ for $\alpha\in C$. Since $A_\alpha=A^*\cap\alpha$ on a stationary set of $\alpha$ in $M$, it follows that $B_\alpha=(\dot A_\alpha)_G=A\cap \alpha$ for stationary many $\alpha$. So $\vec B$ witnesses $\Diamond$ in $M[G]$. 
