Compact directed acyclic word graphs (CDAWGs) [Blumer et al. 1987] are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string $T$ is obtained by merging isomorphic subtrees of the suffix tree [Weiner 1973] of the same string $T$, thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation (insertion, deletion, or substitution) is performed at the left-end of the input string $T$, namely, we are interested in the worst-case increase in the size of the CDAWG after a left-end edit operation. We prove that if $e$ is the number of edges of the CDAWG for string $T$, then the number of new edges added to the CDAWG after a left-end edit operation on $T$ does not exceed $e$. Further, we present a matching lower bound on the sensitivity of CDAWGs for left-end insertions, and almost matching lower bounds for left-end deletions and substitutions. We then generalize our lower-bound instance for left-end insertions to leftward online construction of the CDAWG, and show that it requires $Ω(n^2)$ time for some string of length $n$.