fix(scheduler-targets-alpha): imported lambda function as schedule ta…

…rget throws synth error (#31837)

### Issue # (if applicable)

Closes #29284.

### Reason for this change

Removed the same env check between the Schedule and the Lambda target to allow use of imported Lambda function as target.

### Description of changes

Removed the check that forces the Schedule and the Lambda function to be in the same account and region.

### Description of how you validated changes

- Unit test added to ensure no synth error when using imported Lambda function.
- Integ test added to ensure imported function works. The test ensures the lambda is executed and did what it is supposed to do (i.e. added tag to itself).

### Checklist
- [x] My code adheres to the [CONTRIBUTING GUIDE](https://github.com/aws/aws-cdk/blob/main/CONTRIBUTING.md) and [DESIGN GUIDELINES](https://github.com/aws/aws-cdk/blob/main/docs/DESIGN_GUIDELINES.md)

----

*By submitting this pull request, I confirm that my contribution is made under the terms of the Apache-2.0 license*

packages/@aws-cdk/aws-scheduler-targets-alpha/lib/lambda-invoke.ts

@@ -1,34 +1,23 @@
 import { ISchedule, IScheduleTarget } from '@aws-cdk/aws-scheduler-alpha';
-import { Names } from 'aws-cdk-lib';
 import { IRole } from 'aws-cdk-lib/aws-iam';
 import * as lambda from 'aws-cdk-lib/aws-lambda';
 import { ScheduleTargetBase, ScheduleTargetBaseProps } from './target';
-import { sameEnvDimension } from './util';
 
 /**
  * Use an AWS Lambda function as a target for AWS EventBridge Scheduler.
  */
 export class LambdaInvoke extends ScheduleTargetBase implements IScheduleTarget {
+  private readonly func: lambda.IFunction;
+
   constructor(
-    private readonly func: lambda.IFunction,
-    private readonly props: ScheduleTargetBaseProps,
+    func: lambda.IFunction,
+    props: ScheduleTargetBaseProps,
   ) {
     super(props, func.functionArn);
+    this.func = func;
   }
 
-  protected addTargetActionToRole(schedule: ISchedule, role: IRole): void {
-    if (!sameEnvDimension(this.func.env.region, schedule.env.region)) {
-      throw new Error(`Cannot assign function in region ${this.func.env.region} to the schedule ${Names.nodeUniqueId(schedule.node)} in region ${schedule.env.region}. Both the schedule and the function must be in the same region.`);
-    }
-
-    if (!sameEnvDimension(this.func.env.account, schedule.env.account)) {
-      throw new Error(`Cannot assign function in account ${this.func.env.account} to the schedule ${Names.nodeUniqueId(schedule.node)} in account ${schedule.env.region}. Both the schedule and the function must be in the same account.`);
-    }
-
-    if (this.props.role && !sameEnvDimension(this.props.role.env.account, this.func.env.account)) {
-      throw new Error(`Cannot grant permission to execution role in account ${this.props.role.env.account} to invoke target ${Names.nodeUniqueId(this.func.node)} in account ${this.func.env.account}. Both the target and the execution role must be in the same account.`);
-    }
-
+  protected addTargetActionToRole(_schedule: ISchedule, role: IRole): void {
     this.func.grantInvoke(role);
   }
 }