diff --git a/clap_complete/src/env/mod.rs b/clap_complete/src/env/mod.rs
index 69e0b1d7233..aa5a7964a77 100644
--- a/clap_complete/src/env/mod.rs
+++ b/clap_complete/src/env/mod.rs
@@ -295,6 +295,11 @@ pub trait EnvCompleter {
     /// Write the `buf` the logic needed for calling into `<VAR>=<shell> <cmd> --`, passing needed
     /// arguments to [`EnvCompleter::write_complete`] through the environment.
     ///
+    /// - `var`: see [`CompleteEnv::var`]
+    /// - `name`: an identifier to use in the script
+    /// - `bin`: the binary being completed
+    /// - `completer`: the command to run to generate completions
+    ///
     /// **WARNING:** There are no stability guarantees between the call to
     /// [`EnvCompleter::write_complete`] that this generates and actually calling [`EnvCompleter::write_complete`].
     /// Caching the results of this call may result in invalid or no completions to be generated.
diff --git a/clap_complete/src/env/shells.rs b/clap_complete/src/env/shells.rs
index 1b71c8dd0ee..521cb43e99f 100644
--- a/clap_complete/src/env/shells.rs
+++ b/clap_complete/src/env/shells.rs
@@ -23,8 +23,6 @@ impl EnvCompleter for Bash {
         buf: &mut dyn std::io::Write,
     ) -> Result<(), std::io::Error> {
         let escaped_name = name.replace('-', "_");
-        let mut upper_name = escaped_name.clone();
-        upper_name.make_ascii_uppercase();
 
         let completer =
             shlex::try_quote(completer).unwrap_or(std::borrow::Cow::Borrowed(completer));
@@ -61,7 +59,6 @@ fi
         .replace("NAME", &escaped_name)
         .replace("BIN", bin)
         .replace("COMPLETER", &completer)
-        .replace("UPPER", &upper_name)
         .replace("VAR", var);
 
         writeln!(buf, "{script}")?;
@@ -341,17 +338,18 @@ impl EnvCompleter for Zsh {
     fn write_registration(
         &self,
         var: &str,
-        _name: &str,
+        name: &str,
         bin: &str,
         completer: &str,
         buf: &mut dyn std::io::Write,
     ) -> Result<(), std::io::Error> {
+        let escaped_name = name.replace('-', "_");
         let bin = shlex::try_quote(bin).unwrap_or(std::borrow::Cow::Borrowed(bin));
         let completer =
             shlex::try_quote(completer).unwrap_or(std::borrow::Cow::Borrowed(completer));
 
         let script = r#"#compdef BIN
-function _clap_dynamic_completer() {
+function _clap_dynamic_completer_NAME() {
     local _CLAP_COMPLETE_INDEX=$(expr $CURRENT - 1)
     local _CLAP_IFS=$'\n'
 
@@ -367,7 +365,8 @@ function _clap_dynamic_completer() {
     fi
 }
 
-compdef _clap_dynamic_completer BIN"#
+compdef _clap_dynamic_completer_NAME BIN"#
+            .replace("NAME", &escaped_name)
             .replace("COMPLETER", &completer)
             .replace("BIN", &bin)
             .replace("VAR", var);
diff --git a/clap_complete/tests/snapshots/home/dynamic-env/exhaustive/zsh/zsh/_exhaustive b/clap_complete/tests/snapshots/home/dynamic-env/exhaustive/zsh/zsh/_exhaustive
index b280f7c62e0..e59950fc841 100644
--- a/clap_complete/tests/snapshots/home/dynamic-env/exhaustive/zsh/zsh/_exhaustive
+++ b/clap_complete/tests/snapshots/home/dynamic-env/exhaustive/zsh/zsh/_exhaustive
@@ -1,5 +1,5 @@
 #compdef exhaustive
-function _clap_dynamic_completer() {
+function _clap_dynamic_completer_exhaustive() {
     local _CLAP_COMPLETE_INDEX=$(expr $CURRENT - 1)
     local _CLAP_IFS=$'\n'
 
@@ -15,4 +15,4 @@ function _clap_dynamic_completer() {
     fi
 }
 
-compdef _clap_dynamic_completer exhaustive
+compdef _clap_dynamic_completer_exhaustive exhaustive