Check out example codes for "cognito authentication in AWS using C#". It will help you in understanding the concepts better.

Code Example 1

. . .
using Amazon.CognitoIdentityProvider;
using Amazon.CognitoIdentityProvider.Model;

    public class CognitoUserManager : UserManager<CognitoUser>
    {

        private readonly AmazonCognitoIdentityProviderClient _client = 
            new AmazonCognitoIdentityProviderClient();
        private readonly string _clientId = ConfigurationManager.AppSettings["CLIENT_ID"];
        private readonly string _poolId = ConfigurationManager.AppSettings["USERPOOL_ID"];

        public CognitoUserManager(IUserStore<CognitoUser> store)
            : base(store)
        {
        }

	 . . .

        public override Task<bool> CheckPasswordAsync(CognitoUser user, string password)
        {
            return CheckPasswordAsync(user.UserName, password);
        }

        private async Task<bool> CheckPasswordAsync(string userName, string password)
        {
            try
            {
                var authReq = new AdminInitiateAuthRequest
                {
                    UserPoolId = ConfigurationManager.AppSettings["USERPOOL_ID"],
                    ClientId = ConfigurationManager.AppSettings["CLIENT_ID"],
                    AuthFlow = AuthFlowType.ADMIN_NO_SRP_AUTH
                };
                authReq.AuthParameters.Add("USERNAME", userName);
                authReq.AuthParameters.Add("PASSWORD", password);

                AdminInitiateAuthResponse authResp = await _client.AdminInitiateAuthAsync(authReq);

                return true;
            }
            catch
            {
                return false;
            }
        }
    . . .
    }

Code Example 2

. . .
using Amazon.CognitoIdentityProvider;
using Amazon.CognitoIdentityProvider.Model;

    public class CognitoUserStore : IUserStore<CognitoUser>, 
                                    IUserLockoutStore<CognitoUser, string>, 
                                    IUserTwoFactorStore<CognitoUser, string>
    {
        private readonly AmazonCognitoIdentityProviderClient _client = 
            new AmazonCognitoIdentityProviderClient();
        private readonly string _clientId = ConfigurationManager.AppSettings["CLIENT_ID"];
        private readonly string _poolId = ConfigurationManager.AppSettings["USERPOOL_ID"];

        . . . 
    }

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